NPTEL Syllabus
Advanced Matrix Theory and Linear
Algebra for Engineers - Video course
COURSE OUTLINE
Introduction, Vector Spaces, Solutions of Linear Systems, Important Subspaces
associated with a matrix, Orthogonality, Eigenvalues and Eigenvectors,
Diagonalizable
Matrices, Hermitian Matrices, General Matrices, Jordan
Canonical form (Optional)*, Selected Topics in Applications (Optional)*
http://nptel.iitm.ac.in
Mathematics
COURSE DETAIL
Module
No.
NPTEL
Topic/s
Hours
Coordinators:
1
Introduction:
3
1. First Basic Problem – Systems of Linear equations Matrix Notation – The various questions that arise with
a system of linear eqautions
2. Second Basic Problem – Diagonalization of a square
matrix – The various questions that arise with
diagonalization
2
Vector Spaces
6
1. Vector spaces
2. Subspaces
3. Linear combinations and subspaces spanned by a set
of vectors
4. Linear dependence and Linear independence
5. Spanning Set and Basis
6. Finite dimensional spaces
7. Dimension
3
Solutions of Linear Systems
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
4
Simple systems
Homogeneous and Nonhomogeneous systems
Gaussian elimination
Null Space and Range
Rank and nullity
Consistency conditions in terms of rank
General Solution of a linear system
Elementary Row and Column operations
Row Reduced Form
Triangular Matrix Factorization
Important Subspaces associsted with a matrix
1.
2.
3.
4.
5.
6
Range and Null space
Rank and Nullity
Rank Nullity theorem
Four Fundamental subspaces
Orientation of the four subspaces
4
Prof. Vittal Rao
Centre For Electronics Design and
TechnologyIISc Bangalore
5
Orthogonality
5
1.
2.
3.
4.
5.
6.
7.
8.
Inner product
Inner product Spaces
Cauchy – Schwarz inequality
Norm
Orthogonality
Gram – Schmidt orthonormalization
Orthonormal basis
Expansion in terms of orthonormal basis – Fourier
series
9. Orthogonal complement
10. Decomposition of a vector with respect to a subspace
and its orthogonal complement – Pythagorus Theorem
6
Eigenvalues and Eigenvectors
5
1. What are the ingredients required for diagonalization?
2. Eigenvalue – Eigenvector pairs
3. Where do we look for eigenvalues? – characteristic
equation
4. Algebraic multiplicity
5. Eigenvectors, Eigenspaces and geometric multiplicity
7
Diagonalizable Matrices
5
1. Diagonalization criterion
2. The diagonalizing matrix
3. Cayley-Hamilton theorem, Annihilating polynomials,
Minimal Polynomial
4. Diagonalizability and Minimal polynomial
5. Projections
6. Decomposition of the matrix in terms of projections
8
Hermitian Matrices
5
1. Real symmetric and Hermitian Matrices
2. Properties of eigenvalues and eigenvectors
3. Unitary/Orthoginal Diagonalizbility of Complex
Hermitian/Real Symmetric matrices
4. Spectral Theorem
5. Positive and Negative Definite and Semi definite
matrices
9
General Matrices
5
1. The matrices AAT and ATA
2. Rank, Nullity, Range and Null Space of AAT and ATA
3. Strategy for choosing the basis for the four
fundamental subspaces
4. Singular Values
5. Singular Value Decomposition
6. Pseudoinverse and Optimal solution of a linear system
of equations
7. The Geometry of Pseudoinverse
10
Jordan Cnonical form*
1.
2.
3.
4.
5.
Primary Decomposition Theorem
Nilpotent matrices
Canonical form for a nilpotent matrix
Jordan Canonical Form
Functions of a matrix
5
11
Selected Topics in Applications*
1.
2.
3.
4.
5.
8-10
Optimization and Linear Programming
Network models
Game Theory
Control Theory
Image Compression
A joint venture by IISc and IITs, funded by MHRD, Govt of India
http://nptel.iitm.ac.in